Previous studies of the explanation of NANOGrav 12.5-yr data [1]} with stochastic gravitational wave background from FOPT shows that the NANOGrav result bounds the FOPT at temperature \(T_n\sim \mathcal {O}(1-100)\)  MeV [2]}, [3]}.
This kind of low scale FOPT may occur in the dark sectors weakly interact with the Standard Model [4]}.
Considering three sources of GWs from FOPT (sound wave, MHD turbulence and collision term), we plot the GW spectra and EPTA, PPTA, NANOGrav11 bounds on the stochastic gravitational waves in Fig. REF  and Fig. REF .
The dark U(1) model allow FOPT with a strong phase transition strength in the parameter space of \(M_D\sim MeV\)  and \(g_D\sim \mathcal {O}(0.1)\) . The FOPT strength \(\alpha \)  in \(Z_3\)  model is much smaller with the scalar and pseudoscalar \(M_{s,\chi }\sim \mathcal {O}(10)\)  MeV.
The current NANOGrav 12.5-yr results could be a confinement for FOPT of the dark U(1) when the dark gauge boson masses are of order of MeV. When the \(r_V=\sqrt{3/2}\) , these BM points can account for \((g-2)_\mu \)  anomaly. For the case of dark \(Z_3\)  complex scalar model, in the parameter spaces that can address the \((g-2)_\mu \)  anomaly, the magnitude of GW is too low to be probed by current pulsar timing array experiments.
